If you're in the mood to ruin
Millennium. Frank Black's daughter has to go to the hospital, and while sitting there watching her, he gets an idea that results in catching the killer. His daughter's grave injury has no purpose but to serve as a plot device; it doesn't contribute to character development or affect the overall arc of the story. It's never mentioned again.
The Mentalist. Almost every character introduced in an episode has some relevance to the plot. Whether they turn out to be the killer, a way to catch the killer, or some way for Patrick Jane to prove he's an all-around good guy. One can look at the example of Kimball Cho's girlfriend; the character is introduced early on, but we never see her. The only time we see her is so she can get attacked by a group of thugs and send Cho on a vigilante mission. After that, no mention again.
(Full disclosure: I have not seen the 3rd season.)
One is over a decade old, the other is recent, just to give some perspective.
The thing is, this is a personal gripe. I understand that there's only so much time; in the case of an hour-long drama, it's 40-odd minutes. You don't have time to weave a tale with multiple, unnecessary characters. You have to spend what little time there is on the ones that count. Nevertheless, it annoys me, and where I can, I break the don't need it, cut it rule. I believe it makes the story better, that in order to weave a great story, you need a few "unnecessary" yarns.
The point I'm coming at is that space is limited, and, much of the time, so are the stories. What can be told has to be told within the confines of the opening and closing credits. Ultimately, there is no true end to a story, as the film is merely a snapshot of some point in the characters' lives. After the scene has faded to black, and the credits have rolled, life goes on. That magical wedding becomes a memory as the realities of marriage settle in; you learn to deal with that shocking death. Off-Screen Tales, then, is an exploration of that. A look at the lives of our favourite characters after the camera has long since stopped rolling.